The renormalization of \(Z[J]\)  is obtained by means of the
standard procedure denominated
“renormalized perturbation theory" [1]}. In this scheme, the lagrangian parameters assume their renormalized values and, in order to maintain the validity of the normalization conditions at each order of perturbation theory, local counterterms are introduced, which cancel exactly all the contributions to these parameters which are obtained in the loop expansion. The normalization conditions for the model defined be the action () concern the values of the mass, of the coupling constant and the wave function
normalization. Finally, in order to complete the list of the normalization conditions, one needs to require the absence of a proper vertex which is linear in the field. Let \(\Gamma [\varphi ]\)  be the effective action which corresponds to the sum of the one-particle-irreducible diagrams with external legs represented by \(\varphi (x)\) . In agreement with the structure of the lagrangian (), the additional normalization condition is given by \( \left( \delta \Gamma / \delta \varphi (x) \right) |_{\varphi =0} = 0 \) . Note that, in the case of the \(\phi ^4\)  model, the vanishing of the proper vertices which are
linear and cubic in powers of the fields is a consequence of the symmetry \( \varphi \rightarrow - \varphi \)  which is imposed to the effective action. In the case of gauge fields, the analogue of the condition \( \left( \delta \Gamma / \delta \varphi (x) \right) |_{\varphi =0} = 0 \)  is automatically satisfied.
